U = min(4x, 2x+y)
Your example: x = 15, y = 10
Then, U(15,10) = 40
Question:
Whats py = ? in order to satisfy U(x,y) = 40
Solution:
We assume that we can find py because we assume that the person is maximizing his utility for given prices.
First, you must know that px.x + py.y = budget. You want to know what is the price of the good y that
allows the person to buy many combinations of goods with the same utility U(x,y)=40
Let's think about the conditions under x and y that makes U(x,y)=40:
1)There are three cases were Utility is never 40:
If x < 10, U < 40
if x > 20, U > 40
if x = 10, y<20, U < 40
2)If x = 10, y>20, U(10, y) = 40 always.
We cant use this case to find py. There is no restriction under y that we can use to find py.
3)If 10 <= x <= 20, then we must have 2x + y = 40 in order to make U (x, y) = 40
In this case, 0 <= y <= 20, and we now can make assumptions under py.
Lets solve for this case:
We want to max: px.x + py.y subject to some conditions:
x=>10
x<=20
2x+y=40
So, we have (for px =10): L = 10.x + py.y -l1(2x + y - 40)
deriving:
dL/dx = 0: 10 - l1.2 = 0 -> l1 = 5
dL/dy = 0: py - l1 = 0 -> py = l1 -> py = 5
Biding condition
dL/dl1 = 0: 2x+y-40=0
We find that py = 5 is a condition to make U(x,y)=40, if 10 <= x <= 20, 0 <= y <= 20.
Now, we'll have the following budget, price and quantities in your example:
x=15
y=10
px=10
py=5
py.y + px.x = 5.10 + 10.15 = 200
You can now test and see if py=5 is ok for the budget 200:
Ex1) All these cases U(x,y)=40
For 10 <= x <= 20 and 0 <= y <= 20
x=20 y=0 -> px.x + py.y = 200
x=19 y = 2 -> 10.19 + 5.2 = 200
x=18 y = 4 -> 10.18 + 5.4 = 200
..
x=11 y = 18 -> 10.11 + 5.18 = 110 + 90 = 200
x=10 y = 20 -> 10.10 + 5.20 = 200
Ex2)
For x>20 or x<10, or x=10 and y>20, U is different from 40.
Ex4)
Now, suppose the price was different from py=5, Think that py=20
Then, your budget, for px=10, x=15 and py=20, y=10 would be:
budget: 10.15 + 20.10 = 350
But, if you had 350, you could buy only x goods: x=35 y=0
But in this case, your utility would be U(35,0)= min(140, 70) = 70
And you would be in a better position…
In the other hand, if py = 5, then your budget is 200,
10.x + 5.y = 200
and if you do the same thing, only buys x:
10.20 + 5.0 = 200 -> U=40
See that from the conditions we have described earlier, the only way this consumer could get more utility would be if he get x>20. But since px=10, and the budget is 200, it is impossible. He can only get x=20.
Note that all the cases where py is not 5, you can reorganizer your consumption in order to make more utility than U=40, or maybe you can never get the same utility U=40. But if you could do that, then you would not being rational at first place.. And note that we assumed that the person was maximizing his utility for given prices.